Let's start with the axes.
The vertical axes just counts the number of events that meet some criteria. (And the units even tell you how big a bin is.)
The horizontal axis tells you (in a imprecise and 'round-about way) the criteria: events in which two 'electrons' (actually one electron and one positron; but the distinction doesn't show when using only the particles letter) were detected (which may-or may not have been subject to some kind of pairing requirement—read the caption or the paper), and whose combined mass falls in the bins shown.
Then we have to know what is meant by "combined mass". The invarient mass of a particle or system is $m \equiv (\sqrt{E^2 - (\mathbf{p}c)^2})/c^2$ where $E$ is total energy, $\mathbf{p}$ is the vector momentum, and $c$ is the speed of light. For a system you get $E_{sys}$ and $\mathbf{p}_{sys}$ by adding the values for each part. So we add the energy of the two 'electrons' and add their three-momenta and then compute $m_{ee}$. (The detector measures the energy and momentum of particles, so we're down to processed output at that point.)
The dots with error bars represent the actual number of events detected and the error bars represent an estimate of how close we might expect the same dot to land if we ran the whole experiment again (from statistical arguments).
So what else do we see on the figure?
There are two lines plotted. Down near the very bottom is a narrow band of blue with a line on top of it, and much higher is another line running more or less through the data. Both represent calculations of the number of events that would be expected from some well understood theory. The theory results not corresponding to what you want to show are sometimes called "backgrounds" The blue band is the QCD background comes from well understood interaction between the quarks and gluons that make up the protons that are being collided in this experiment (ATLAS is a LHC experiment). The white (or clear) band represents a calculation of the signal expected from the weak decay of a Z-boson (so it is not a background because it is what we are looking for); you don't always see that on discovery plots unless the discovery was expected as with the Higgs). Because the wite band sits on top of the blue band the line at the top of the white band is the combined theoretical background plus signal.
What you are suppose to notice is that the data and the background-plus-signal line match quite closely. (Important thing about error bars: they are plotted at the one-sigma level so about one in three points should have its error bar not quite get to the line and about one in twenty should miss by a lot—that's statistics for you.)
If this was a discovery announcement they'd still have some work to do in terms of angular distributions and estimating the spin and parity of the progenitor.
